2. SEPTEMBER 1ST: LOGIC
3
2. September 1st: Logic
2.1
.
Prove De Morgan’s laws.
Answer:
Recall that these say
¬
(
p
∨
q
)
⇐⇒
(
¬
p
)
∧
(
¬
q
)
¬
(
p
∧
q
)
⇐⇒
(
¬
p
)
∨
(
¬
q
)
.
We can take care of these with two truth tables:
p
q
¬
p
¬
q
p
∨
q
¬
(
p
∨
q
)
(
¬
p
)
∧
(
¬
q
)
T
T
F
F
T
F
F
T
F
F
T
T
F
F
F
T
T
F
T
F
F
F
F
T
T
F
T
T
proves the first law, and
p
q
¬
p
¬
q
p
∧
q
¬
(
p
∧
q
)
(
¬
p
)
∨
(
¬
q
)
T
T
F
F
T
F
F
T
F
F
T
F
T
T
F
T
T
F
F
T
T
F
F
T
T
F
T
T
proves the second.
Note that the second law actually follows from the first (or conversely, if you
prefer), so one table would su
ffi
ce. To see this, assume you have proved the first law;
negate both sides to get
p
∨
q
⇐⇒
¬
((
¬
p
)
∧
(
¬
q
))
;
then apply this tautology to
p
=
¬
r
and
q
=
¬
s
:
(
¬
r
)
∨
(
¬
s
)
⇐⇒
¬
(
r
∧
s
)
.
Of course
r
and
s
are just names for arbitrary statements, so the last tautology we
obtained is just a restatement of the second law.
2.2
.
Assume that
L
(
x
) means
‘
x
is a lion’;
C
(
x
) means
‘
x
is a cow’;
R
(
x
) means
‘
x
is red’;
G
(
x
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 Fall '11
 Aluffi
 Logic, Universal quantification, Lion

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